The low Mach number limit for the isentropic Euler system with axisymmetric initial data

E-ISSN： 1475-3030|12|2|335-389

ISSN： 1474-7480

Source： Journal of the Institute of Mathematics of Jussieu, Vol.12, Iss.2, 2013-04, pp. : 335-389

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Abstract

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is ${H}^{s} $ with $s\gt \frac{5}{2} $ . Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the lifespan ${T}_{\varepsilon } $ of the solutions is bounded below by $\log \log \log \frac{1}{\varepsilon } $ , where $\varepsilon $ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system when the parameter $\varepsilon $ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity ${ B}_{2, 1}^{\frac{5}{2} } $ . This case turns out to be more subtle because of at least two features. The first one is related to the Beale–Kato–Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate ${ L}_{T}^{1} {L}^{\infty } $ for the acoustic parts $(\nabla {\Delta }^{- 1} \mathrm{div} \hspace{0.167em} {v}_{\varepsilon } , {c}_{\varepsilon } )$ : it scales in the space variables like the space of the initial data.

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